4.4.10  
Brannon and Boyce (baby 3rd ed.)

solution

lambda = -2;
omega = 2 Sqrt[31];

y[t_]=-(1/(4 omega)) E^(lambda t) Sin[omega t]

Simplify[y''[t]+128 y[t]+4 y'[t]]

y[0]
y'[0]

first pass through equilibrium

basicplot=Show[Plot[y[t],{t,0,2.5},PlotRange->All],ParametricPlot[{Pi/omega,t},{t,0,0.01},PlotStyle->Dashed],Plot[0.01/12,{t,0,2.5},PlotStyle->Dashed],PlotRange->All]

small amplitude

rought=Log[150/Sqrt[31]]/2
N[rought]

farplot=Plot[{y[t],-0.01/12,0.01/12},{t,rought-0.5,rought+0.5}]

Show[farplot,ParametricPlot[{rought,t},{t,-0.001,0.001},PlotStyle->Dashed]]

exactt=FindRoot[y[tfind]==0.01/12, {tfind, 1.62}][[1,2]]

Show[farplot,ParametricPlot[{exactt,t},{t,-0.001,0.001},PlotStyle->Dashed]]